Optimal. Leaf size=28 \[ \frac {1}{x}+x-\frac {25}{\log ^2\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \]
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Rubi [F] time = 7.99, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {50 x-50 x^2+e^{4 x+x^2} \left (-200 x^2-100 x^3\right )+\left (x-x^3+e^{4 x+x^2} \left (1-x^2\right )+e^3 \left (-1+x^2\right )+\left (-1+x^2\right ) \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )}{\left (e^3 x^2-e^{4 x+x^2} x^2-x^3+x^2 \log (x)\right ) \log ^3\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {50 x-50 x^2-100 e^{x (4+x)} x^2 (2+x)-\left (-1+x^2\right ) \left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}{x^2 \left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx\\ &=\int \left (\frac {50 \left (1-\left (1+4 e^3\right ) x+4 \left (1-\frac {e^3}{2}\right ) x^2+2 x^3-4 x \log (x)-2 x^2 \log (x)\right )}{x \left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}+\frac {200 x^2+100 x^3-\log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )+x^2 \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}{x^2 \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}\right ) \, dx\\ &=50 \int \frac {1-\left (1+4 e^3\right ) x+4 \left (1-\frac {e^3}{2}\right ) x^2+2 x^3-4 x \log (x)-2 x^2 \log (x)}{x \left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+\int \frac {200 x^2+100 x^3-\log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )+x^2 \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}{x^2 \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx\\ &=50 \int \left (-\frac {1}{x \left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}-\frac {2 x^2}{\left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}+\frac {2 x \log (x)}{\left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}-\frac {1+4 e^3}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}-\frac {2 \left (-2+e^3\right ) x}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}-\frac {4 \log (x)}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}\right ) \, dx+\int \left (1-\frac {1}{x^2}+\frac {100 (2+x)}{\log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}\right ) \, dx\\ &=\frac {1}{x}+x-50 \int \frac {1}{x \left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+100 \int \frac {2+x}{\log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx-100 \int \frac {x^2}{\left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+100 \int \frac {x \log (x)}{\left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx-200 \int \frac {\log (x)}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+\left (100 \left (2-e^3\right )\right ) \int \frac {x}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx-\left (50 \left (1+4 e^3\right )\right ) \int \frac {1}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx\\ &=\frac {1}{x}+x-50 \int \frac {1}{x \left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+100 \int \left (\frac {2}{\log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}+\frac {x}{\log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )}\right ) \, dx-100 \int \frac {x^2}{\left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+100 \int \frac {x \log (x)}{\left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx-200 \int \frac {\log (x)}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+\left (100 \left (2-e^3\right )\right ) \int \frac {x}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx-\left (50 \left (1+4 e^3\right )\right ) \int \frac {1}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx\\ &=\frac {1}{x}+x-50 \int \frac {1}{x \left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+100 \int \frac {x}{\log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx-100 \int \frac {x^2}{\left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+100 \int \frac {x \log (x)}{\left (-e^3+e^{x (4+x)}+x-\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+200 \int \frac {1}{\log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx-200 \int \frac {\log (x)}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx+\left (100 \left (2-e^3\right )\right ) \int \frac {x}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx-\left (50 \left (1+4 e^3\right )\right ) \int \frac {1}{\left (e^3-e^{x (4+x)}-x+\log (x)\right ) \log ^3\left (-e^3+e^{x (4+x)}+x-\log (x)\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.10, size = 30, normalized size = 1.07 \begin {gather*} \frac {1}{x}+x-\frac {25}{\log ^2\left (-e^3+e^{4 x+x^2}+x-\log (x)\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 56, normalized size = 2.00 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \relax (x)\right )^{2} - 25 \, x}{x \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \relax (x)\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.53, size = 75, normalized size = 2.68 \begin {gather*} \frac {x^{2} \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \relax (x)\right )^{2} + \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \relax (x)\right )^{2} - 25 \, x}{x \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \relax (x)\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 32, normalized size = 1.14
method | result | size |
risch | \(\frac {x^{2}+1}{x}-\frac {25}{\ln \left (x +{\mathrm e}^{\left (4+x \right ) x}-{\mathrm e}^{3}-\ln \relax (x )\right )^{2}}\) | \(32\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.53, size = 56, normalized size = 2.00 \begin {gather*} \frac {{\left (x^{2} + 1\right )} \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \relax (x)\right )^{2} - 25 \, x}{x \log \left (x - e^{3} + e^{\left (x^{2} + 4 \, x\right )} - \log \relax (x)\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 29, normalized size = 1.04 \begin {gather*} x-\frac {25}{{\ln \left (x-{\mathrm {e}}^3-\ln \relax (x)+{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{x^2}\right )}^2}+\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.57, size = 26, normalized size = 0.93 \begin {gather*} x - \frac {25}{\log {\left (x + e^{x^{2} + 4 x} - \log {\relax (x )} - e^{3} \right )}^{2}} + \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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